A moving interface method for dynamic kinetic–fluid coupling

This paper is a continuation of earlier work [P. Degond, S. Jin, L. Mieussens, A smooth transition between kinetic and hydrodynamic equations, Journal of Computational Physics 209 (2005) 665–694] in which we presented an automatic domain decomposition method for the solution of gas dynamics problems which require a localized resolution of the kinetic scale. The basic idea is to couple the macroscopic hydrodynamics model and the microscopic kinetic model through a buffer zone in which both equations are solved. Discontinuities or sharp gradients of the solution are responsible for locally strong departures to local equilibrium which require the resolution of the kinetic model. The buffer zone is drawn around the kinetic region by introducing a cut-off function, which takes values between zero and one and which is identically zero in the fluid zone and one in the kinetic zone. In the present paper, we specifically consider the possibility of moving the kinetic region or creating new kinetic regions, by evolving the cut-off function with respect to time. We present algorithms which perform this task by taking into account indicators which characterize the non-equilibrium state of the gas. The method is shown to be highly flexible as it relies on the time evolution of the buffer cut-off function rather than on the geometric definition of a moving interface which requires remeshing, by contrast to many previous methods. Numerical examples are presented which validate the method and demonstrate its performances.     

A Moving Mesh Method for Kinetic/Hydrodynamic Coupling

This paper deals with the application of a moving mesh method for kinetic/hydrodynamic coupling model in two dimensions. With some criteria, the domain is dynamically decomposed into three parts: kinetic regions where fluids are far from equilibrium, hydrodynamic regions where fluids are near thermodynamical equilibrium and buffer regions which are used as a smooth transition. The Boltzmann-BGK equation is solved in kinetic regions, while Euler equations in hydrodynamic regions and both equations in buffer regions. By a well defined monitor function, our moving mesh method smoothly concentrate the mesh grids to the regions containing rapid variation of the solutions. In each moving mesh step, the solutions are conservatively updated to the new mesh and the cut-off function is rebuilt first to consist with the region decomposition after the mesh motion. In such a framework, the evolution of the hybrid model and the moving mesh procedure can be implemented independently, therefore keep the advantages of both approaches. Numerical examples are presented to demonstrate the efficiency of the method.     

An h-adaptive mesh method for Boltzmann-BGK/hydrodynamics coupling

We introduce a coupled method for hydrodynamic and kinetic equations on 2-dimensional h-adaptive meshes. We adopt the Euler equations with a fast kinetic solver in the region near thermodynamical equilibrium, while use the Boltzmann-BGK equation in kinetic regions where fluids are far from equilibrium. A buffer zone is created around the kinetic regions, on which a gradually varying numerical flux is adopted. Based on the property of a continuously discretized cut-off function which describes how the flux varies, the coupling will be conservative. In order for the conservative 2-dimensional specularly reflective boundary condition to be implemented conveniently, the discrete Maxwellian is approximated by a high order continuous formula with improved accuracy on a disc instead of on a square domain. The h-adaptive method can work smoothly with a time-split numerical scheme. Through h-adaptation, the cell number is greatly reduced. This method is particularly suitable for problems with hydrodynamics breakdown on only a small part of the whole domain, so that the total efficiency of the algorithm can be greatly improved. Three numerical examples are presented to validate the proposed method and demonstrate its efficiency.     

Kinetic boundary layers in gas mixtures: Systems described by nonlinearly coupled kinetic and hydrodynamic equations and applications to droplet condensation and evaporation

We consider a mixture of heavy vapor molecules and a light carrier gas surrounding a liquid droplet. The vapor is described by a variant of the Klein-Kramers equation, a kinetic equation for Brownian particles moving in a spatially inhomogeneous background; the gas is described by the Navier-Stokes equations; the droplet acts as a heat source due to the released heat of condensation. The exchange of momentum and energy between the constituents of the mixture is taken into account by force terms in the kinetic equation and source terms in the Navier-Stokes equations. These are chosen to obtain maximal agreement with the irreversible thermodynamics of a gas mixture. The structure of the kinetic boundary layer around the sphere is then determined from the self-consistent solution of this set of coupled equations with appropriate boundary conditions at the surface of the sphere. For this purpose the kinetic equation is rewritten as a set of coupled moment equations. A complete set of solutions of these moment equations is constructed by numerical integration inward from the region far away from the droplet, where the background inhomogeneities are small. A technique developed in an earlier paper is used to deal with the severe numerical instability of the moment equations. The solutions so obtained for given temperature and pressure profiles in the gas are then combined linearly in such a way that they obey the boundary conditions at the droplet surface; from this solution source terms for the Navier-Stokes equation of the gas are constructed and used to determine improved temperature and pressure profiles for the background gas. For not too large temperature differences between the droplet and the gas at infinity, self-consistency is reached after a few iterations. The method is applied to the condensation of droplets from a supersaturated vapor, where small but significant corrections to an earlier, not fully consistent version of the theory are found, as well as to strong evaporation of droplets under the influence of an external heat source, where corrections of up to 40 % are obtained.     

Description of the inner state and kinetics of Rydberg atoms based on the kinetic Boltzmann equation for Rydberg electrons

The inner characteristics and kinetics of Rydberg atoms (RAs) excited selectively over energy in a buffer gas are considered using the kinetic equation for a classical distribution function of Rydberg electrons (REs). The distribution of REs over coordinates and velocities in a moving RA is found in the general case. In a moving RA, the effect of “ blowing off” an electron cloud by a buffer gas is substantial. In this case, however, the average values of the kinetic and potential energies of REs weakly deviate from those predicted by the virial theorem. The latent and macroscopic polarizations of the medium caused by the blowing-off effect are predicted. The macroscopic polarization appears upon velocity-selective excitation of RAs and produces the bias current, which transforms to a usual electric current when the integrity of the RA is lost due to the blowing-off effect. The calculated “ electron” contribution to the transport frequency of collisions of the RA with buffer gas atoms proved to be small compared to that from the ion core.     

A reinterpretation of dense gas kinetic theory

In dense gas kinetic theory it is standard to express all reduced distribution functions as functionals of the singlet distribution function. Since the singlet distribution function includes aspects of correlated particles as well as describing the properties of freely moving particles, it is here argued that these aspects should more clearly be distinguished and that it is the distribution function for free particles that is the prime object in terms of which dense gas kinetic theory should be expressed. The standard equations of dense gas kinetic theory are rewritten from this point of view and the advantages of doing so are discussed.     

Analysis of Fokker-Planck type stochastic equations with variable boundary conditions in an elementary process of collisional ionization

The diffusion model of ionization of Rydberg atoms in single collisions is formulated. The solution of kinetic equations describing nonstationary diffusion processes is discussed. Methods of analytical calculation of the effective times for the development of ionization in phase regions with moving boundaries are presented. An analytical method, going back to Weisskopf, for determining the corresponding cross sections for impact ionization in the adiabatic approximation is described. A numerical scheme for solving problems of stochastic movement of a Rydberg electron inside a Coulomb condensation of levels under conditions of time-variable ionization boundaries and region of development of stochastic instability is developed.     

Kinetic modeling of a slow flow CW CO2 laser

A kinetic model for analysis of the slow-flow CW-discharge CO₂ laser with diffusion cooling has been developed in which the gas temperature is obtained from energy balance equations. The method is based on the numerical solution of a set of nonlinear differential equations for vibrational kinetics. The numerical predictions from the model are compared with some experimental results and a good agreement is obtained.     

Green function theory of dynamic conductivity

The transport processes can be discussed either by kinetic equation method or by correlation function method. Using the former, linear transport equations are developed for the study of dynamic conductivity of a quantum imperfect gas employing a resolution of BBGKY hierarchy using Green functions. From this transport equation a modified form of Kubo (correlation function) formula is obtained to show the equivalence between the two methods. This equivalence may be used for the justification of the concept of adiabatic switching of the field. The simple formula derived, gives the conductivity in terms of one-particle Green function, unlike the usual discussions which express it in higher order Green functions.     

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证明了用能量原理推导出的关于圆柱等离子体局部模的Suydam不稳定性,对应的动能变化量趋于无穷大,故其增长率趋于零。通过对Hain-Lust方程求解知道,Suydam不稳定性的真实物理意义是,它对应于一系列低模数的广域不稳定模,其增长率随模数加大而减小。实际上,任何扭曲模分析都一定要正确处理共振面附近的求解问题。提出了不满足Suydam判据时外扭曲模的一个求解方法。     

A fast iterative model for discrete velocity calculations on triangular grids

A fast synthetic type iterative model is proposed to speed up the slow convergence of discrete velocity algorithms for solving linear kinetic equations on triangular lattices. The efficiency of the scheme is verified both theoretically by a discrete Fourier stability analysis and computationally by solving a rarefied gas flow problem. The stability analysis of the discrete kinetic equations yields the spectral radius of the typical and the proposed iterative algorithms and reveal the drastically improved performance of the latter one for any grid resolution. This is the first time that stability analysis of the full discrete kinetic equations related to rarefied gas theory is formulated, providing the detailed dependency of the iteration scheme on the discretization parameters in the phase space. The corresponding characteristics of the model deduced by solving numerically the rarefied gas flow through a duct with triangular cross section are in complete agreement with the theoretical findings. The proposed approach may open a way for fast computation of rarefied gas flows on complex geometries in the whole range of gas rarefaction including the hydrodynamic regime.     

Energy spectrum of fast atoms in a dark cathode space

A method is proposed for solving the kinetic equation for fast atoms moving in a proper gas. We find the Green's function which permits computation of the energy spectrum of fast atoms for an arbitrary source function in the volume of a proper gas or an amorphous body having plane boundaries. The energy spectrum of fast atoms is computed in a dark cathode space. The solution obtained is used to compute the contribution of fast atoms to atomization of the cathode surface.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 62–67, August, 1987.     

Two linearization procedures for the Boltzmann equation in ak=0 Robertson-Walker space-time

It is the aim of this paper to describe two different linearization procedures for the Boltzmann equation in ak=0 Robertson-Walker space-time. These procedures are discussed with a view to obtaining an asymptotic form of the Boltzmann equation for the late stages of cosmic expansion where the behavior appropriate to a nonrelativistic gas is encountered. Using the asymptotic kinetic equations, a necessary and sufficient condition is formulated under which every small perturbation of the equilibrium distribution function, either classical or relativistic, decays to zero as time goes on. The same condition can be extracted from each of two linearization procedures, and in this sense a comparison is made of these approaches which reveals mutual agreement. Also, applying an asymptotic theory of the Einstein-Boltzmann system, we show that the final state of a gas is dust (i.e., a fluid with zero temperature and pressure). Comparison with the predictions of the Eckart fluid model is briefly presented.     

Kinetic Models for Granular Flow

The generalization of the Boltzmann and Enskog kinetic equations to allow inelastic collisions provides a basis for studies of granular media at a fundamental level. For elastic collisions the significant technical challenges presented in solving these equations have been circumvented by the use of corresponding model kinetic equations. The objective here is to discuss the formulation of model kinetic equations for the case of inelastic collisions. To illustrate the qualitative changes resulting from inelastic collisions the dynamics of a heavy particle in a gas of much lighter particles is considered first. The Boltzmann–Lorentz equation is reduced to a Fokker–Planck equation and its exact solution is obtained. Qualitative differences from the elastic case arise primarily from the cooling of the surrounding gas. The excitations, or physical spectrum, are no longer determined simply from the Fokker–Planck operator, but rather from a related operator incorporating the cooling effects. Nevertheless, it is shown that a diffusion mode dominates for long times just as in the elastic case. From the spectral analysis of the Fokker–Planck equation an associated kinetic model is obtained. In appropriate dimensionless variables it has the same form as the BGK kinetic model for elastic collisions, known to be an accurate representation of the Fokker–Planck equation. On the basis of these considerations, a kinetic model for the Boltzmann equation is derived. The exact solution for states near the homogeneous cooling state is obtained and the transport properties are discussed, including the relaxation toward hydrodynamics. As a second application of this model, it is shown that the exact solution for uniform shear flow arbitrarily far from equilibrium can be obtained from the corresponding known solution for elastic collisions. Finally, the kinetic model for the dense fluid Enskog equation is described.     

The influence of temperature gradients on the kinetic boundary layer problem for a condensing droplet

We extend an earlier method for solving kinetic boundary layer problems to the case of particles moving in aspatially inhomogeneous background. The method is developed for a gas mixture containing a supersaturated vapor and a light carrier gas from which a small droplet condenses. The release of heat of condensation causes a temperature difference between droplet and gas in the quasistationary state; the kinetic equation describing the vapor is the stationary Klein-Kramers equation for Brownian particles diffusing in a temperature gradient. By means of an expansion in Burnett functions, this equation is transformed into a set of coupled algebrodifferential equations. By numerical integration we construct fundamental solutions of this equation that are subsequently combined linearly to fulfill appropriate mesoscopic boundary conditions for particles leaving the droplet surface. In view of the intrinsic numerical instability of the system of equations, a novel procedure is developed to remove the admixture of fast growing solutions to the solutions of interest. The procedure is tested for a few model problems and then applied to a slightly simplified condensation problem with parameters corresponding to the condensation of mercury in a background of neon. The effects of thermal gradients and thermodiffusion on the growth rate of the droplet are small (of the order of 1%), but well outside of the margin of error of the method.     

碲溶剂法生长碲镉汞晶体的数值模拟

提出了碲溶剂法在稳态条件下生长碲镉汞晶体的理论模型.该模型利用与时间相关的一维物质扩散方程组，熔区自由边界通过相图来确定.采用有限差分法完成了组分x=0.2的长波碲镉汞晶体生长过程的数值模拟.讨论了液相区温度场分布、加热器移动速度、液相区长度和生长界面温度对生长碲镉汞晶体轴向组分的影响.模拟结果与实验结果相符. 关键词：     

基于动理论模型的一维等离子体电磁波传输特性分析

本文采用了动理论模型对电磁波在等离子体中的传播特性进行了研究.建立了与该模型相关的麦克斯韦-玻尔兹曼(MB)方程组,并采用FDTD方法加以求解,得到了等离子体区域内的粒子分布函数及空间中的电场分布.此外,本文还计算了电磁波入射到等离子体平板上的反射及透射系数,并将数值计算结果与解析解结果进行了比较,验证了该方法的正确性.     

Molecular dynamics of interacting kinks. I

A system consisting of a large number (up to 40 000) of kinks and antikinks moving under attractive interactions, which are annihilated on contacting each other, is studied using the method of molecular dynamics computer simulation. The average distance between neighboring kinks increases logarithmically in time after a short initial transient period. The size distribution function of domains between neighboring kinks is also computed and found to develop a characteristic cut-off structure. Interpretation of the results in terms of simple kinetic models is given. The results are compared with the recent neutron scattering experiments on layered antiferromagnets by Ikeda.     

Anomalous Diffusion Limit Induced on a Kinetic Equation

Using a compactness argument based on the velocity averaging lemma of Golse et al., it is shown that the limiting behavior of a kinetic (linearized BGK) gas model confined between two plates with Maxwell boundary conditions, when the distance between the plates goes to zero, under a suitable anomalous scaling, is diffusive. We do not require the use of central limit theorems as in the method of Börgers et al.